For first person shooters (mix of first introduced and popularised):

Doom: started and popularised the genre. Also started and popularised rasterized 3d graphics for gaming (though the game itself was still 2d). Also first fps multiplayer and modding

Quake: various game modes (Deathmatch, capture the flag), as well as being the first true 3d fps. Popularised multiplayer and modding.

Team fortress (quake mod): Different specialist characters.

Goldeneye 64: popularized multilayer console fps, taught character size can be a significant advantage/disadvantage, depending on if you got Oddjob or Jaws.

Half-life: started horror fps genre, (mostly) seemless world

CS: customizable loadouts instead of search for guns each time you spawn, more game modes

UT: AI bots

Perfect dark: secondary fire for weapons

Deus ex: rpg fps

Halo: finally figured out a decent controller control scheme (one stick looks, one moves, button for grenades rather than needing to select grenade from list of guns). First fps I remember vehicles in, too.

Battlefield: large scale multiplayer

Socom: fps game that isn't first person, online console multiplayer

Call of duty: using gun sights to aim

Far cry: open world fps

Doom 3: used lighting (or lack thereof) to bring fps horror to a new level.

Crisis: famous for pushing hardware and people caring more about the benchmark results than the game itself (I tried the second one, it was ok but I didn't really get into it)

Call of duty: zombies (and other alternate game modes), kill steaks, online progression (unlocking guns and attachments as you level, prestige levels)

HL2/portal: brought physics and its involvement in fps games to a new level

TF2: f2p, microtransactions (though not predatory or p2w so the game isn't remembered for this)

Borderlands: loot-based fps rpg

Metro 2033: fps survival

Halo reach: custom maps

Destiny: MMORPG FPS

Overwatch: hero-based, and hero roles (dps, tank, healer)

Pub bg: battle Royale